Another relevant body of literature pertains to interval arithmetic [1]}, [2]}, [3]}, [4]}, [5]}, which  has been successfully applied to problems in numerical analysis, set estimation, motion planning, etc. Specifically, inclusion functions and variations thereof (e.g., natural,
centered-form and mixed-form inclusions) that are based on interval arithmetic can be directly related to decomposition functions and can similarly be used for over-approximation of the ranges/image sets of functions and for state observer designs [6]}, [7]}, [8]}, [9]}, [10]}, [5]}. In addition, various properties of inclusion functions have been studied such as their convergence rates and the subdivision principle [2]}. Subsequent studies further introduced refinements of interval over-approximations and set inversion algorithms that can incorporate
new sources of information about the system such as state constraints, measurements/observations, manufactured redundant variables, second-order derivatives, etc., e.g., in [1]}, [6]}, [7]}, [8]}, [9]},  which can be also beneficial for decomposition function-based constrained reachability and set-valued estimation problems that we consider in this  work.
